It will be their seventh consecutive appearance in the last eight of the competition as the Red Devils bounced back from their weekend Premier League woes.
In a highly lacklustre first half, Manchester United dominated possession, but struggled to create anything tangible. The biggest chance of the half came from Victor Lindelof, who headed into the woodwork.
Even in the second half, both teams looked uninspired. However, West Ham came close to getting a late win at the death, but Manchester United did well to hold on and take the game to extra time.
West Ham looked more threatening after going behind, but did not do enough to get the equaliser; their focus will now shift back to the Premier League.
Meanwhile, Ole Gunnar Solskjaer's side are now three wins away from winning their first FA Cup title in five years. On that note, let us have a look at the Manchester United player ratings.
Dean Henderson 6/10
Dean Henderson was in goal for Manchester United in place of the under-fire David de Gea. But he had nothing to do in the first 45 minutes. He remained unthreatened in the second half, as the visitors' first and only shot on target came in the second half of extra time.
Aaron Wan-Bissaka 6.5/10
Aaron Wan-Bissaka was very solid defensively, completing four tackles and four interceptions and snuffing out the West Ham attack down his flank. He, however, offered nothing in attack, and was removed at the end of normal time.
Victor Lindelof 7/10
The Manchester United defender came closest to opening the scoring with a header, which was tipped wide by Fabianski onto the post. In an assured performance, Victor Lindelof calmly played his side out of danger on many occasions.
Harry Maguire 7/10
The Manchester United captain was utterly dominant in the air, as he won long balls for his side. Harry Maguire won a game-high ten aerial duels and also made five clearances to ensure Manchester United's progress to the next round.
Alex Telles 7/10
Alex Telles was the most lively of the Manchester United fullbacks, as he drove forward and put in useful balls into the box.
The Brazilian created four chances and completed five crosses before he came off for Luke Shaw in extra time.
Nemanja Matic 6.5/10
Nemanja Matic kept play ticking for Manchester United and had the most touches in his 73 minutes on the pitch. He was very tidy in possession and looked to break opposition lines with his passes.
Fred sent long balls forward to create something for his side, but they were either overhit or unused by his teammates.
He was often exploited due to his overzealous pressing, but did well to recover on each occasion. Fred ended the game with the most touches and passes completed in the game.
Donny Van de Beek 5.5/10
The Dutchman was often situated in deeper regions than he probably should have been and struggled to create anything for his side. Donny Van de Beek made way for Bruno Fernandes in the 73rd minute.
Mason Greenwood 6/10
The teenager was the tidiest of all Manchester United attackers; he was solid on the ball, winning all but one of his ground duels.
Mason Greenwood was well defended by West Ham, and couldn't make much of an impact on the game.
Anthony Martial 5.5/10
The Frenchman was missing for most of the game and looked short on ideas. Anthony Martial, however, created the chance which Rashford fluffed at the start of the second half.
Marcus Rashford 6/10
The Manchester United number 10 ran himself into traffic in a bid to take opposition defenders on.
Marcus Rashford was also gifted a glorious chance to open the scoring early in the second half, but could only hit his shot straight at the keeper.
He did provide the assist for Manchester United's goal, but it was a subpar performance by the Englishman.
Player ratings for Manchester United Substitutes
Bruno Fernandes 6/10
Often Manchester United's go-to man in difficult situations, Bruno Fernandes was brought on for Van de Beek to help unlock the Hammers defence.
He, however, struggled to make an impact on the game and looked out of pace.
Scott McTominay 8/10
Scott McTominay was brought in as a more goal-oriented alternative to Matic, and he proved to be just that.
He arrived just in time to meet Marcus Rashford's lay-off and smack it first time into the bottom corner to take his goal tally to seven in all competitions this season.
Edinson Cavani 6/10
The Manchester United forward was brought in when the scores were level . However, Edinson Cavani was well defended by the London side and could not do much.
Brandon Williams 6/10
He defended well when he was tested. However, Brandon Williams could have done better blocking West Ham's crosses, as a bulk of them came through his flank in the dying minutes of the game.
Luke Shaw 6/10
The Manchester United defender was lively after coming on, playing his teammates into promising positions.